Donald Cary Williams

The published work of Donald Williams (1899–1983) ranges
across a broad spectrum in philosophy, but his importance as a
philosopher rests in large measure on four major
achievements. Firstly, in a period when the role of philosophy was
being diminished and trivialized, he persisted with a traditional
style of philosophizing. Although it remained unfashionable
throughout most of his active years, he held to the classic program of
Western philosophy: to explain and defend our capacity to attain
knowledge (so far as that reaches), in the light of that to present a
well-reasoned account of the most general and fundamental features of
Reality, and on that basis to propose a wise scheme for the good
life. His unswerving allegiance to this project, keeping it alive in
dark days, has had its reward. At least in Anglophone circles, these
classic metaphysical, epistemic and ethical issues are once more
firmly on the agenda.

Williams’ own contributions belong to the epistemic and
metaphysical aspects of the project rather than its ethical side
(except for a brief defense of ethics as ultimately “pure
postulate” (1933b)). So, secondly, as an account of our
world he proposed an empirically informed materialistic
naturalism. Among its distinctive features, he champions a
4-dimensional space-time which accords equal status to Past, Present,
and Future.

Thirdly, this cosmology is embedded in an original and powerful
theory of the nature of properties, the thesis that properties are
themselves particulars rather than universals. This Trope theory, as it
has come to be known, following Williams’ own usage, has become
one of the standard alternatives in the Problem of Universals.

And in the fourth place, in the teeth of the prevailing skepticism
on this topic, he developed an original and boldly positive
justification of inductive inference.

Donald Williams was born on 28 May 1899 in Crow’s Landing,
California, at that time a strongly rural district, and died in
Fallbrook, also in his beloved California, and also at that time far
from cities, on 16 January 1983. His father was Joseph Cary
Williams, who seems to have been a jack of all countrymen’s
trades; his mother Lula Crow, a local farmer’s daughter. Donald
was the first in his family to pursue an academic education. After
studies in English Literature at Occidental College (BA 1922), he went
to Harvard for his Masters, this time in Philosophy (AM 1924). He
then undertook further graduate study in philosophy, first at the
University of California at Berkeley (1925–27), then at Harvard, where
he took his PhD in 1928.

Also in 1928 he married Katherine Pressly Adams, from Lamar,
Colorado, whom he had met at Berkeley, where she was something of a
pioneer—a woman graduate student in psychology. In time,
there were two sons to the marriage. The couple spent a year in Europe
in 1928–29 (“immersing himself in Husserl’s Phenomenology
to the point of immunization”, Firth, Nozick and Quine 1983).
Then Donald began his life’s work as a Professor of Philosophy,
first spending ten years at the University of California, Los Angeles,
and then from 1939 until his retirement in 1967, at Harvard.

Throughout his long and distinguished career in philosophy he retained
a down-to-earth realism and naturalism in metaphysics, and a
conservative outlook on moral and political issues, characteristic of
his origins. A stocky, genial, and cheerful man, he found neither the
content nor the validation of ethics to be problematic. Although
originally a student of literature, whose first publication was a book
of poetry, he had not the slightest tincture of literary or academic
bohemianism. He was among the very least alienated academics of
his generation; which is not surprising, as his career was indeed one
version of The American Dream.

The traditional ambition of philosophy, in epistemology and
metaphysics, is to provide a systematic account of the extent and
reliability of our knowledge, and on that basis, to provide a synoptic
and well-based account of the main features of Reality. When
Williams was in his prime, this ambition was largely repudiated as
inappropriate or unattainable, and a much more modest role for
philosophy was proposed. In setting forth his own position, in the
preface to the collection of his selected essays (1966, p.viii), he
lists some of these fashionable philosophies from the mid-twentieth
century:

…logical positivism, logical behaviorism,
operationalism, instrumentalism, the miscellaneous American
linguisticisms, the English varieties of Wittgensteinism, the
Existentialisms, and Zen Buddhism…

Each of these is, in
its own way, a gospel of relaxation. They all propose that, in place of
the struggle to uncover how things are, careful descriptions of how
things appear will suffice. None is ambitious enough to set about
constructing a positive and systematic epistemology and
metaphysics.

Undeterred by this spirit of the age, Williams continued to insist
that philosophical issues are real and large questions, having genuine
answers. Conceptual analysis, concentration on phenomenological
description, or exploration of the vagaries of language, may have their
(subordinate) place, but to elevate them to a central position is an
evasion of philosophy’s main task.

Still worse was the suggestion that philosophical questions are mere
surface expressions of a philosopher’s underlying
psycho-pathology. The claim of Morris Lazerowitz to that effect,
suggesting that Bradley’s Absolute Idealism was no more than an
intellectual’s poorly expressed death wish, or that
McTaggart’s argument against the reality of time a panic fear of
change, he met with a stern rebuke (1959, pp 133–56).

He set forth a Realist philosophy on traditional empiricist
principles: “He thought that practically everything was right out
there where it belonged” (Firth, Nozick and Quine 1983). He
maintained that while all knowledge of fact rests on perceptual
experience, it is not limited to the perceptually given, but can be
extended beyond that by legitimate inference (1934a). In this way his
Realism can develop the breadth and depth required to do justice to all
the scientific techniques which so far surpass mere perception.

Williams’s empiricism extended to philosophy itself. He
challenged the prevailing orthodoxy that philosophy is a purely a
priori discipline. He emphasized the provisional character of much
philosophizing, and the striking absence of knock-down arguments in
philosophic controversy (“Having Ideas In The Head” 1966,
189–211).

Following his own prescription for an affirmative and constructive
philosophy, Williams worked steadily towards the development of his own
distinctive position in metaphysics. He introduced the useful division
of the subject into Speculative Cosmology, which deals with the basic
elements making up the world we live in, such as matter, mind, and
force, together with the relations between them, and Analytic Ontology,
which explores the fundamental categories, such as Substance and
Property, and how they relate to one another. Speculative
Cosmology, in particular, needs to be open to developments in the
fundamental sciences, and so needs to be seen as always provisional and
a posteriori. Analytic Ontology, the exploration of the
categories of being, is a more purely reflective discipline aiming to
elucidate the range of different elements in any universe.

To begin with Speculative Cosmology, Williams’s position has
three main features.

It is Naturalistic. The natural world of space, time, and matter,
with all its constituents, is a Reality in its own right. Contrary to
all Idealist metaphysics, this world, except for the finite minds that
are to found within it, is independent of any knowing mind. Moreover,
it is the only world. There are no divinities or supernatural powers
beyond the realm of Nature (“Naturalism and the Nature of
Things” 1966, pp 212–238). Even mathematical realities
belong in the natural world: numbers—at least natural
ones—as abstractions from clusters, and geometrical objects as
abstractions from space. The philosophy of mathematics was an aspect
of his position that was never fully worked through.

Second, his position is not only Naturalist but also Materialist.
This Materialism is not of the rather crude kind that supposes that
every reality is composed of a solid, crunchy substance, the stuff that
makes up the atoms of Greek speculation. Any spatio-temporal
reality, whether an ‘insubstantial’ property, such as a
color, or something as abstract as a relation, such as
farther-away-than, or faster-than, so long as it takes its place as a
spatio-temporal element at home in the world of physics, is accepted as
part of this one great spatio-temporal world. Williams’s
Materialism is thus one which can accept whatever physical theory
posits as the most plausible foundation for the natural sciences,
provided that it specifies a world which develops according to natural
law, without any teleological (final) causes. This openness to
developments in physics is what makes his an Empirical Realism.

The Mental is accommodated in the same way. Although Mind is
unquestionably real, mental facts are as spatio-temporally located as
any others (“The Existence of Consciousness”, “Mind
as a Matter of Fact” 1966, pp 23–40, 239–261). The Mental is not
an independent realm parallel to and equal to the Material, but rather
a tiny, rather insignificant fragment of Being, dependent upon, even if
not reducible to, the physical or biological nature of living beings.
Williams has a capacious conception of the Material—his
position could perhaps be better described as Spatio-Temporal
Naturalism.

Thirdly, Williams’s metaphysics is 4-dimensional. Or,
more precisely, given the development of multi-dimensional string
theories since his time, it takes Time to be a dimension in the same
way that Space has dimensions. The first step is to insist, against
Aristotle and his followers, that statements about the future, no less
than those concerning the present and the past, are timelessly true or
false. They need not await the event they refer to, in order to gain a
truth value (“The Sea Fight Tomorrow” 1966, pp 262–288.)
This encourages the further view that the facts that underpin truths
about the future are (timelessly) Real. All points in time are
(timelessly) Real, as are all points on any dimension of our familiar
Space. Whatever account is to be given of Change, it does not consist
in items gaining or losing Reality. This stance receives powerful
support from physical theory. Williams embraced and argued for
the conception of Time as a fourth dimension introduced by
Minkowski’s ‘Block Universe’ interpretation of
Einstein’s Theory of Special Relativity. A consequence of this is
that the experience of the flow or passage of Time must be some sort of
illusion. Williams embraced that consequence, and argued for it
in a celebrated paper (‘The Myth of Passage’, 1951).

Apart from his Materialistic or Spatio-Temporal Naturalism,
Williams’s major contributions to metaphysics lie in the realm of
ontology, and concern the fundamental constituents of Being. His
key proposal is that properties are indeed real—in fact Reality
consists in nothing but properties—but that these properties are not
Universals, as commonly supposed, but particulars with unique
spatio-temporal locations. The structure of Reality comprises a
single fundamental category, Abstract Particulars, or
“tropes”. Tropes are particular cases of general
characteristics. A general characteristic, or Universal, such as
redness or roundness, can occur in any one of indefinitely many
instances. Williams’ focus was on the particular case of
red which occurs as the color, for example, of a particular rose at a
specific location in space and time, or the particular case of
circularity presented by some particular coin in my hand on a single,
particular occasion. These tropes are as particular, and as
grounded in place and time, as the more familiar objects, the rose and
the coin, to which they belong.

These tropes are the building blocks of the world. In his analogy,
they provide ‘the Alphabet of Being’ from which the
entities belonging to more complex categories—objects,
properties, relations, events—can be constructed, rather as
words and sentences can be built using the letters of the
alphabet. Familiar objects such as shoes and ships
and lumps of sealing wax, and their parts as revealed by empirical
scientific investigation, such as crystals, molecules and atoms, are
concrete particulars or things. In Williams’s scheme,
each of these consists in a compresentcluster of
tropes—the particular thing’s particular shape, size,
temperature, and consistency, its translucency, or acidity, or positive
charge, and so on. All the multitude of different tropes that
comprise some single complex particular do so by virtue of their
sharing one and the same place, or sequence of places, in
Space-Time. That is what ‘compresent’ means. There is
no inner substratum or individuator to hold all the tropes
together. The tropes are individuals in their own right, and do
not inhere in any thing-like particular. So Williams’s view
is a No-Substance theory, or, otherwise described, a theory in which
each trope is itself a simple Humean substance, capable of independent
existence.

Universal properties and quantities such as acidity and velocity,
which are common to many objects, are not beings in their own right,
but resemblance classes of individual tropes. If two
objects match in color, both being red, for example, the tropes of
color belonging to each are separate tropes, both being members of the
class of similar color tropes which constitutes Redness.

Relations are treated along the same lines. If London is Larger-than
Edinburgh, and Dublin Larger-than Belfast, we have two instances of the
Larger-than relation, two relational tropes. And they, along with
countless other cases, all belong to the resemblance class whose
members are all and only the cases of Larger-than. This account denies
that, literally speaking, there is any single entity which is
simultaneously fully present in two different cases of the same color,
or temperature, or whatever. So it is a No-Universals view, and
often described as a version of Nominalism. This is
understandable, but it is better to confine the term
‘Nominalism’ to the denial of the reality of properties at
all. So far from denying properties, on Williams’s theory the
entire world consists in nothing but tropes, which are properties
construed as particulars. So his position is better described not as
Nominalist but as Particularist.

Tropes provide an elegant and economical base for an ontology.
Unlike almost all others, this one of Williams rests on just one basic
category, which can be used in the construction not just of things and
their properties and relations, but of further categories, such as
events and processes. Events are changes in just which tropes are to be
found in a given location, the replacement of one trope by another.
Processes are sequences of such changes. Trope theory is well placed to
furnish an attractive analysis of causality, as involving power tropes
that govern and drive the transformations to be found in events and
processes.

It can also be of use in other areas of philosophy, for example in
valuation theory, where the existence of many tropes, rather than one
single unitary reality, can explain our sometimes divided attitudes
toward what, on a Substance ontology, we would regard as the same
thing. Something can be good in some respects (tropes), but not in
others. To view the manifest world as comprising, for the most
part, clusters of compresent tropes makes explicit the complexity of
the realities with which we are ordinarily in contact.

Many philosophers have admitted tropes into their scheme of
things: Aristotle, Locke, Spinoza and Leibniz, for example. What
is distinctive in Williams is not that tropes are admitted as a
category, but as the only fundamental category, a trope-based
form of what Schaffer calls “property primitivism.”
(Schaffer 2003, 125). All else is constructed out of tropes, including
concrete particulars and general properties, whereas tropes themselves
are not constructed out of anything else, for example, out of a
substance, a universal and the relation of exemplification. Not
surprisingly, various aspects of Williams’ trope primitivism have
been subjected to serious philosophical challenges either directly or
indirectly.

Some philosophers reject all forms of property primitivism,
including that of Williams, on the grounds that properties cannot serve
as the only independent elements of being. They lack the
requisite independence, the capacity of existing in any combination
with “wholly distinct existences.” There are two ways
for this objection to go. (1) Armstrong takes the ostensive fact that
properties must be had by objects to establish the dependence of
properties on non-property particulars, substrata, and, thus, the
falsity of property primitivism (Armstrong 1989, 115). (2)
Alternately, one can take the apparent fact that there can be no
properties that are not clustered with other properties to show that
properties cannot play this role.

In response to (1), one can challenge the assumption that
if properties must be had by objects, then they are not capable of
independent existence. Ross, for example, suggests that
properties might both be capable of independent existence—including
existing without substrata—and not be capable of existing without
being the property of something. (2006, 104). The dependence of
properties on objects is compatible with property primitivism if one
adopts a bundle theory of concrete particulars. In that case,
properties are always had by objects since they are always found in
bundles, even if only a bundle of one, but exist without substrata.

As for (2), some philosophers reject the requirement that
if properties are the only ultimate constituents of reality, then they
must be capable of existing in isolation from all other wholly distinct
properties (Simons 1994; Denkel 1997). Properties can be both the only
ultimate constituents of reality and inter-dependent. This
response requires the rejection of the Humean principle that the basic
independent units of being can exist in any combination,
including unaccompanied (Schaffer 2003, 126). A very different
response to (2) rejects the necessity of trope clustering altogether
(Williams 1966, 97; Campbell 1981, 479; Schaffer 2003).

Plausible
though it be, however, that a color or a shape cannot exist by itself,
I think we have to reject the notion of a standard of concreteness.
… (Williams 1966, 97)

At best, it is a contingent
matter that there are no tropes that are not compresent with any other
tropes—for example, a mass trope on its own. (Campbell goes so
far as to suggest that there is reason to think that there are actual
cases of free-floating tropes (1981, 479)).

Even if the trope primitivist can get around these quite
general objections, there remains the most serious objection to
Williams’ brand of trope primitivism, an objection that is
specific to Williams’ conception of tropes, depending on more
than the assumption that tropes are properties. The charge is that a
Williamsonian trope is not genuinely simple, but complex embracing, at
least, an element that furnishes the nature or content of the trope,
and an element providing its particularity (Hochberg 1988; Armstrong
2005; Moreland 1985; Ehring 2011). In short, Williamsonian tropes
are constructed out of something else, making them incompatible with
trope primitivism.

This objection is based on two assumptions. First,
under Williams’ conception, the nature of a trope is a
non-reducible, intrinsic matter that is not determined by relations to
anything else, including resemblance relations to other tropes or
memberships in various natural classes of tropes. And, second, anything
that stands in more than one arbitrarily different relation, each of
which is grounded intrinsically in that entity, must be complex since
that entity will have intrinsic “aspects” that are not identical to
each other. The two relevant relations are numerical difference
from other tropes and resemblance to other tropes, each of which is
grounded intrinsically in the trope relata under the Williams
conception. Hence, there are “intrinsic aspects” of
each trope that are not identical to each other, a
particularity-generating component and a nature-generating
component.

In response to this “complexity” objection,
Campbell claims that the distinction between a trope’s nature and
its particularity is merely a “formal” distinction, a
product of different levels of abstraction, and not a real distinction
between different components of a trope. One should no more
distinguish a particularity-component from a nature-component of a
trope than distinguish components of warmth and orangeness in an orange
trope:

To recognize the case of orange as warm is not to find a
new feature in it, but to treat it more abstractly, less specifically,
than in recognising it as a case of orange. (Campbell 1990, 56–7)

Alternately, Ehring suggests that we grant this objection,
but preserve trope simplicity by switching to a non-Williamsonian
conception of tropes, according to which a trope’s nature is
determined by its memberships in various natural classes of tropes
rather than intrinsically, thereby sidestepping one of the assumptions
operative in the objection (2011).

Coming under criticism as well is Williams’
resemblance-based account of general characteristics and property
agreement. According to Williams, a fully determinate general
color characteristic—say, the shade of red that characterizes this
shirt and this chair—-is just the set of all tropes that exactly
resemble the red trope of this shirt. Different objects of that
“same” shade of red each possess a different trope from
this set of exactly similar red tropes. But this analysis seems to
generate an infinite regress. If trope t1 is related to
trope t2 by resemblance trope r1, t2
is related to trope t3 by resemblance trope r2,
and t3 is related to t1 by resemblance trope
r3, then these resemblance tropes will also resemble each
other, giving rise to further resemblance tropes, and so on. To stop
this vicious regress, the objection continues, resemblance must be
taken to be a universal (Daly 1997, 150).

One response to this objection tries to stop the regress
before it starts by denying that there are any resemblance
trope-relations holding between tropes. In particular, the trope
theorist might follow Oliver’s advise

to avoid saying that
when two tropes are exactly similar …, there exists a
relation-trope of exact similarity … holding between the two
tropes. (1996, 37)

There are no resemblance-tropes
corresponding to these resemblance predicates. Another response
denies that resemblance relations mark an addition to our ontology and,
hence, there can be no regress of resemblance relations. Campbell, for
example, argues that the successive resemblance relations in the
regress are nothing over and above the non-relational tropes that
ultimately ground these relations. Since resemblance is an
internal relation, it supervenes on these ground-level non-resemblance
tropes, but supervenient “additions” are not real additions
to one’s ontology (Campbell 1990, 37).

There is also a whole host of objections in the literature
to Williams’ trope bundle theory. According to Williams, concrete
particulars are not substrata instantiating various properties.
They are wholly constructed out of tropes, forming bundles of tropes,
the trope constituents of which are pairwise tied together by a
compresence relation. Compresence, in turn, is collocation for
Williams, “the unique congress in the same volume.”
(In order to allow for non-spatial objects, Williams grants the
possibility of “locations” in systems analogous to space
(1966, 79).) One immediate worry, raised by Campbell, is the
possibility that there may be cases of overlapping objects
demonstrating that collocation is not sufficient for compresence (1990,
footnote 5, 175). In response, the trope bundle theorist can opt
for the view that compresence is non-reducible. However, even with this
revision there remain significant objections concerning the possibility
of accidental properties, the possibility of change, and an apparent
vicious regress of compresence relations.

Bundle theory has been charged with ruling out the
possibility of accidental properties in concrete particulars.
Bundles of properties have all of their constituent properties
essentially. Objects generally do not. This chair could have been
blue instead of red, but the bundle of properties that characterize the
chair could not have failed to include that red property. One way
around this objection is proposed by O’Leary-Hawthorne and Cover:
combine bundle theory (although for them properties are universals)
with a specific account of modality, a counterpart semantics for
statements about ordinary particulars (1998). What makes it true
that a particular object o could have had different properties is that
a non-identical counterpart to o, n, in another possible world has
different properties than does o. As long as there is a possible
world in which there is a object-bundle, n, that is non-identical
counterpart to o and differs from o with respect to its properties,
then o could have differed in just that way.

Simons suggests a very different approach. He
proposes to replace unstructured bundles with “nucleus
theory.” An object consists of an inner core of essential tropes
and an outer band of accidental tropes, but no non-property substratum
(1994).

In like manner, bundle theory has been charged with ruling
out the possibility of change in objects. The same bundle complex
cannot be composed of one set of property at one time, but a different
set at a different time. Concrete objects, on the other hand, can
and do change. In response, it has been suggested that this objection
loses its force if bundle theory is combined with a
four-dimensionalist account of object persistence (Casullo 1988; see
also Ehring 2011). For example, if ordinary objects are spacetime worms
made up of appropriately related instantaneous temporal parts that are
themselves complete bundles of compresent tropes, then change can be
read as a matter of having different temporal parts that differ in
their constitutive properties. Another response to the
change-is-impossible objection is to adopt Simons’ “nucleus
theory” in place of traditional bundle theory. Nucleus
theory seems to allow for change in the outer band of accidental
properties (Simons 1994).

Trope bundle theory has also been accused of giving rise to
Bradley-style vicious regress. According to trope bundle theory,
for an object o to exist, its tropes must be mutually compresent.
However, it would seem, for trope t1 to be compresent with
trope t2, they must be linked by a compresence trope, say,
c1. But, the existence of t1,
t2, and c1 is insufficient to make it the case
that t1 and t2 are compresent since these tropes
could each be parts of different, non-overlapping bundles. So for
t1 and t2 to be linked by compresence trope
c1, c1 must be compresent with t1 by
way of a further compresence relation, say c2 (and with
t2 by, say, c3) and so on, giving rise to either
a vicious, or at least, uneconomical regress (Maurin 2010, 315).
In response, one can try to break the link between the relevant
predicates and tropes. Oliver suggests that the trope theorist
should reject the assumption that there are any relation-tropes
corresponding to the predicate “….is compresent
with…” even though that predicate has some true
applications (1996, 37). This response might be indirectly supported by
reference to Lewis’s claim that it is an impossible task to give
an analysis of all predications since any analysis will bring into play
a new predication, itself requiring analysis (Lewis 1983,
353).

A second, quite different response is modeled on
Armstrong’s view that “instantiation” is a
“tie,” not a relation (since “the thisness and nature are
incapable of existing apart”), and, hence, it is not subject to a
relation regress. (1978, 109). The idea is that the union of compresent
tropes is too intimate to speak of a relation between them since the
tropes of the same object could not have existed apart from each
other. This response, however, requires more than generic
dependencies—for example, that this specific mass trope requires the
existence of some solidity trope or other—since generic dependencies
would not guarantee that the specific mass and solidity tropes, say, in
this particular bundle could not have existed without being
compresent.

Maurin provides an alternative response that grants the
existence of compresence relation-tropes, but denies the regress on the
grounds that relation-tropes, including compresence relations,
necessitate the existence of that which they relate. There is no
need for further compresence relations holding between a
compresence-relation and its terms (2002, 164). A fourth response, from
Ehring, suggests that this regress can be stopped once it is recognized
that compresence is a self-relating relation, a relation that can take
itself as a relatum. The supposed infinite regress for the bundle
theorist involves an unending series of compresence tropes,
c1, c2, …, and cn, but
the series, c1, c2, …, and cn
is taken to be infinite because it is assumed that each “additional”
compresence trope is not identical to the immediate preceding
compresence trope in the series. However, if compresence is a
“self-relating” relation, this assumption may be false
(2011).

Finally, there is an objection to the very notion of a
trope and, hence, to the foundations of trope primitivism (and,
perhaps, to any ontology that includes tropes). The idea is that
if properties are tropes, then exactly similar properties can be
swapped across objects, but there is no such
possibility.

If the redness of this rose is exactly similar to but
numerically distinct from the redness of that rose, then the redness of
this rose could have been the redness of that rose and vice
versa. But this is not really a possibility and, thus, properties
are not tropes. (Armstrong 1989, 131–132)

A similar argument is based on the possibility of tropes swapping
positions in space (Campbell 1990, 71). “Property
swapping,” it is claimed, is an unreal possibility since property
swapping would make no difference to the world. (Note that the
cross-object version of this objection cannot get off the ground if
tropes are not transferable between objects, a view that is found in
(Martin 1980), although Martin rejects trope primitivism since he
posits substrata in addition to tropes).

One response to the no-swapping
objection, given by Campbell and Labossiere, rejects the assumption
that trope swaps would make no difference to the world. For
example, although the effects of “swapped” situations would
be exactly similar in nature, those effects would differ in their
causes (Campbell 1990, 72; Labossiere 1993, 262). Schaffer, on
the other hand, denies that the trope theory is automatically committed
to the possibility of trope swapping (2001). If tropes are
individuated by times/locations and a counterpart theory of modality is
right, then trope swapping is ruled out:

The redness which would be
here has exactly the same inter- and intraworld resemblance relations
as the redness which actually is here, and the same distance relations,
and hence it is a better counterpart than the redness which would be
there. (Schaffer 2001, 253).

What is clear is that Williams’ trope ontology remains at the
center of a vibrant and ongoing debate, counting as a serious option
among a small field of contenders. His brand of trope primitivism
is certainly of more than merely historical interest.

Williams has also left a wider, if less plainly manifest legacy as a
metaphysician. For a discussion of his influence on David Lewis, and
through him on later thinkers, see Fisher 2015.

The Problem of Induction is the problem of vindicating as rational
our unavoidable need to generalize beyond our current evidence to
comparable cases that we have not yet observed, or that never will be
observed. Without inductive inferences of this kind, not only all
science but all meaningful conduct of everyday life is paralyzed.
Indeed, Williams prefaced his philosophical treatment with a
declaration of the evils of inductive skepticism in eroding rational
standards in general, even in politics:

In the political sphere,
the haphazard echoes of inductive skepticism which reach the
liberal’s ear deprive him of any rational right to champion
liberalism, and account already as much as anything for the flabbiness
of liberal resistance to dogmatic encroachments from the left or the
right. (Williams 1947, pp 15–20)

David Hume, in the
eighteenth century, had shown that all such inferences must involve
risk: no matter how certain our premises, no inductively reached
conclusion can have the same degree of certainty. Hume went
further, and held that inductive reasoning provides no rational support
whatever for its conclusions. This is Hume’s famous inductive
scepticism.

Williams was almost alone in his time in holding not only that the
problem does admit of a solution, but in presenting a novel solution of
his own. To do this, he needed to argue against Hume, and Hume’s
twentieth century successors Bertrand Russell and Karl Popper, who had
declared the problem insoluble, and also against contemporaries such as
P. F. Strawson and Paul Edwards, who had claimed that there was no real
problem at all (Russell 1912, Chapter 6; Popper 1959; Edwards 1949, pp
141–163; Strawson 1952, pp 248–263). Williams tackled the
problem head-on. In The Ground of Induction (1947) he makes
original use of results already established in probability theory,
whose significance for the problem of induction he was the first to
appreciate. He treats inductive inference as a special case of the
problem of validating sampling techniques. Among any population, that
is, any class of similar items, there will be a definite proportion
having any possible characteristic. For example, among the population
of penguins, 100% will be birds, about 50% will be female, some 10%,
perhaps, will be Emperor penguins, some 35%, perhaps, will be more than
seven years old, and so on. This is the complexion of the
population, with regard to femaleness, or whatever.

Now in the pure mathematics of the relations between samples and
populations, Jacob Bernoulli had in the 18th century established the
remarkable fact that, for populations of any large size whatever (say
above 2500), the vast majority of samples of 2500 or more closely match
in complexion the population from which they are drawn. In the case of
our penguins, for example, if the population contains 35% aged 7 years
or more, well over nine tenths of samples of 2500 penguins will contain
between 33% and 37% aged 7 years or more. This is a necessary, purely
mathematical, fact. In the language of statistics, the vast
majority of reasonably sized samples are representative of
their population, that is, closely resemble it in complexion.

Bernoulli’s result enables us to infer from the complexion of
a population to the complexion of most reasonably-sized samples taken
from it, since the complexion of most samples closely resembles the
complexion of the population. Williams’s originality was this: he
noticed that resemblance is symmetrical. If we can
prove, as Bernoulli did, that most samples resemble the population from
which they are drawn, then conversely the population’s complexion
resembles the complexion of most of the samples.

This brings us to the problem of induction. What our observations of
the natural world provide us with can be regarded as samples from
larger populations. For example the penguins we have observed up to
this point provide us with a sample of the wider population of all
penguins, at all times, whether observed or not. What can we infer
about this wider population from the sample we have? That, in all
probability, the population’s complexion is close to that of the
sample. We may of course, have an atypical sample before us.
But with samples of more than 2500, well over 90% represent the
population fairly closely, so the odds are against it.

Thus Williams assimilates the problem of induction to an application
of the statistical syllogism (also called direct inference or the
proportional syllogism). A standard syllogism concerns complexions of
100%, and has a determinate conclusion: if all S are P, and
this present item is an S, then it must be P. A
statistical syllogism deals with complexions of less than 100%, and its
conclusion is not definite but only probable: for example, if 95%
of S are P, this present S is probably P. Some
logicians claim that the probability in question is exactly 0.95, but
Williams does not need to rely on that additional claim. It is enough
that the probability be high. Applying the statistical syllogism to
Williams’ reversal of the Bernoulli result, we have: 95% of
reasonably-sized samples are closely representative of their
population, so the sample we have, provided there are no grounds to
think otherwise, is probably one of them. On that basis, we are
rationally entitled to infer that the population probably closely
matches the sample, whose complexion is known to us. The inference is
only probable. Induction cannot deliver certainty. In any given case,
it is abstractly possible that our sample may be a misleading,
unrepresentative one. But to expect an inference from the observed to
the unobserved to yield certainty is to expect the impossible.

Williams’ treatment of induction created quite a stir when it
appeared, but attracted criticisms of varying power from commentators
wedded to more defeatist attitudes, and was eclipsed by the Popperian
strategy of replacing an epistemology of confirmation with one that
focused on refutation. It thus exercised less influence than it
deserves. It is a closely reasoned, deductively argued defense of
the rationality of inductive inference, well meriting continued
attention.

Its reliance on a priori reasoning (as opposed to any contingent
principles such as the “uniformity of nature” or the action
of laws of nature) means that it should hold in all possible worlds.
Williams’ argument would thus be easily defeated by the
exhibition of a possible world in which inductive reasoning did not
work (in the sense of mostly yielding false conclusions). However,
critics of Williams have not offered such a possible non-inductive
world. Chaotic worlds are not non-inductive (as the induction from
chaos to more chaos is correct in them), while a world with an
anti-inductive demon, who falsifies, say, most of the inductive
inferences I make, is not clearly non-inductive either (since although
most of my inductions have false conclusions, it does not
follow that most inductions in general have false conclusions).

Marc Lange (2011) does however propose a counterexample, arising
from the “purely formal” nature of Williams’
argument. Should it not apply equally to “grue” as to
“green”? Objects are grue if they are green up to
some future point in time, and blue thereafter. The problem is to
show that our sample, to date, of green things is not a sample of
things actually grue.

Stove (1986, 131–144) argued that his more specific version of
Williams’s argument (see below) was not subject to the objection, and
that the failure of induction in the case of ‘grue’ showed
that inductive logic was not purely formal—but then neither was
deductive logic.

Other criticisms arise from the suggestion that the proportional
syllogism in general is not a justified form of inference without some
assumption of randomness. Any proportional syllogism (with exact
numbers) is of the form

The proportion of Fs that are G is r.

ais F.

So, the probability that a is G is r.

(Or, if we let B be the proposition that the proportion
of Fs that are G is r, and we
let p(h | e) be the conditional
probability of hypothesis h on evidence e, then we
can express the above proportional syllogism in the language of
probability as follows: p(Ga | Fa
& B) = r.)

Do we not need to assume that a is chosen
“randomly”, in the sense that all Fs have an
equal chance of being chosen? Otherwise, how do we know that
a is not chosen with some bias, which would make its
probability of being G different from r?

Defenders of the proportional syllogism (McGrew 2003; Campbell and
Franklin 2004) argue that no assumption of randomness is needed. Any
information about bias would indeed change the probability, but that is
a trivial fact about any argument. An argument infers from given
premises to a given conclusion; a different argument, with a different
force, moves from some other (additional) premises to that conclusion.
Given just that the vast majority of airline flights land safely, I can
have rational confidence that my flight will land safely, even though
there are any number of other possible premises (such as that I have
just seen the wheels fall off) that would change the probability if I
added them to the argument. The fact that the probability of the
conclusion on other evidence would be different is no reason
to change the probability assessment on the given evidence.

Similar reasoning applies to any other property that a may
have (or, in the case of the Williams argument, that the sample may
have), such as having been observed or being in the past. If there is
some positive reason to think that property relevant to the conclusion
(that a is G), that reason needs to be explained; if
not, there is no reason to believe it affects the argument and the
original probability given by the proportional syllogism stands.

Serious criticisms specific to Williams’s argument have been
based on claims that the proportional syllogism, though correct in
general, has been misapplied by Williams. Any proportional
syllogism,

The proportion of Fs that are G is r.

a is F.

So, the probability that a is G is r,

is subject to the objection that, in the case at hand, there is
actually further information about the Fs that is relevant to
the conclusion Ga. For example, in

The proportion of candidates who will be appointed to the board of
Albert Smith Corp is 10%.

Albert Smith Jr is a candidate.

So, the probability that Albert Smith Jr will be appointed is 10%,

it is arguable that information is hidden in the proper names that
is favorably relevant to the younger Smith’s chance of success.
The question then is whether the same could happen with the
proportional syllogism in Williams’ argument for induction. Maher
(1996) argues that there is such a problem. In this version of
Williams’ argument:

The proportion of large samples whose complexion approximately
matches the population is over 95%.

S is a large sample.

So, the probability that the complexion of S matches the population
is over 95%,

is there, typically, any further information about the sample S that
is relevant to whether it matches the population? The potentially
relevant information one has is the proportion in S of the attribute to
be predicted (blue, or whatever it might be). Can that be relevant to
matching?

It certainly can be relevant. For example, if the proportion of blue
items in the sample is 100%, it suggests that the population proportion
is close to 100%; that is positively relevant to matching since samples
of near-homogeneous populations are more likely to match. (For example,
if the population proportion is 100%, all samples match the
population.) Conversely, fewer samples match the population when the
population proportion is near one half.

David Stove (1986), in defending Williams’ argument, proposed
to avoid this problem by taking a more particular case of the argument
which would not be subject to the objection. Stove proposed:

If F is the class of ravens, G the class of black things,
S a sample of 3020 ravens, r = 0.95, and
‘match’ means having proportion within 3% of the
population proportion, then it is evident that ‘The proportion
of Fs that are Gs is
x’ is not substantially unfavorable to matching (for all x).

For even in the worst case, when the proportion of black things is
one half, it is still true that the vast majority of samples match the
population (Stove 1986, 66–75).

Stove’s reply emphasizes how strong the mathematical truth about
matching of samples is: it is not merely that, for any population size
and proportion, one can find a sample size and degree of match such
that samples of that size mostly match; it is the much stronger result
that one can fix the sample size and degree of match beforehand,
without needing knowledge of the population size and proportion. For
example, the vast majority of 3020-size samples match to within
3%—irrespective of the population size (provided of course that
it is larger than 3020) and the proportion of objects with the
characteristic under investigation..

Maher also objects that the attribute itself, such as
“blue”, might be a priori relevant to its
proportion in the sample and hence to matching. For example, if
“blue” is one of a large range of possible colors,
then a priori it is unlikely that an individual is blue and
unlikely that a sample will have many blue items. As with any Bayesian
reasoning, a prior probability close to zero (or one) requires a lot
of evidence to overcome; in this case, we would conclude that a sample
with a high proportion of blue items was most likely a coincidence,
and the posterior probability of the sample matching the population
would still not be high.

Scott Campbell (2001), in reply to Maher, argues that priors do not
dominate observations in the way Maher suggests. By analogy, suppose
that, while blindfolded, I throw a dart at a dartboard. I am told that
99 of the 100 spots on the board are the same color, and that there are
145 choices of color for the spots. After throwing, I observe just the
spot I have hit and find it is blue. Then (in the absence of further
information), the chance is very high that almost all the other spots
are blue. The prior improbability of blue does not prevent that. In the
same way, the fact that the vast majority of samples match the
population gives good reason to suppose that the observed sample does
too, irrespective of any prior information of the kind Maher
advances.

Williams’ defense of induction thus has resources to supply
answers to the criticisms that have been made of it. It remains the
most objectivist and ambitious justification of induction.

1959, “Philosophy and Psychoanalysis,”
in Psychoanalysis, Scientific Method, and Philosophy; a
Symposium, S. Hook (ed.), New York: New York
University Press. pp. 157–79.
[Williams 1959 available online]

The Donald Williams papers, containing a substantial quantity of
unpublished material, are in the archives of Harvard University. They
include most of the text of a treatise on Logic, in the tradition of
J. M. Keynes, expounding the various species of judgment and
inference, rather than the systematic axiomatizations of deductive
inference, and the metalogic, which superseded that tradition.

McGrew, Timothy, 2003, “Direct Inference and the Problem of
Induction,” in Probability is the Very Guide of Life: the
philosophical uses of chance, H. E. Kyburg and M. Thalos (eds.),
Peru, IL: Open Court, pp 33–60.